Determine whether a function is smooth?

Question

Suppose we have 4 functions $f,g,h,j$ all of them are functions from $R^n$ to $R$, in particular, $f,g$ are smooth. I’m curious, when given following condition: $$fh+gj=0$$can we claim that $h$ and $j$ are also smooth?

Answer 1

Assuming that $g$ does not cancel,

$$j=-\frac fgh$$

solves the equation, so $h,j$ have no reason to be smooth. Not even continuous.

Answer 2

Counterexample: Let $f(x) = x$ and $g(x) = x^2$. Then we could choose $h(x) = \frac{1}{x}$ and $j(x) = -\frac{1}{x^2}$ neither of which are smooth (or even defined) at $x=0$. To be extra clear, we can extend the definition of either $h$ or $j$ to make $h(0) = a$ and $j(0) = b$ for some points $a$ and $b$, but no extension of these functions make them continuous.